I understand that the graphs of $\log(x)$ and $\ln(x)$ both have derivatives (changes in slope) that follow the pattern of:
$$\frac{d}{dx}\log_{b}x= \frac{1}{(x\ln(b))}$$
However, depending on the source, I have seen different definitions for the integral of $\frac{1}{x}$:
$$\int\frac{1}{x}dx=\ln| x |+C$$
$$\int\frac{1}{x}dx=\log| x |+C$$
$$\int\frac{1}{x}dx=\ln x+C$$
$$\int\frac{1}{x}dx=\log x +C$$
I believe that only the top two definitions are close to being valid, and I also think that $\ln| x |+C$ is the only correct answer, based on the formula for the derivative given above, and the fact that $ln(e) = 1$. Is that incorrect?
Can the answer to this be shown graphically as well as algebraically?